1 Preface

The goal of this course is to cover the basics of analysis in a form suitable for
engineers. The aim is to prepare students for advanced courses in differential
equations, stochastics, optimization and related topics.

The students are assumed to have some maturity: they should be familiar with
linear algebra and ordinary calculus. The exercises are provided with three
aims:

to push the student to learn by doing;

to help him/her learn to present results in a clear, lucid style;

to provide examples and facts that are often useful by themselves.

The course starts with the primitive notions and a review of the properties of real
numbers. Then, ideas of closeness, distance, convergence, and continuity are
developed by way of metric spaces. These notions are then used to develop the
derivative and its properties. Next, we cover the theory of integration, and its
basic tools, in abstract measure spaces. Finally, fixed point theory, integral
equations, Fourier analysis, and convergence of certain algorithms will be
covered.

There will be weekly assignments and a final exam. The assignments will count
for 75% of the final grade and the final exam will count for the other
25%.

2 Outline

Sets and Functions

Sets, set operations, real line, infimum and supremum

Functions, injections and surjections, positivity, etc.

Sequences, limits, convergence of sequences, series

Cardinality, countability, uncountable sets

Metric Spaces

Metrics, norms (Manhattan Metric, l2-norm, weighted l2-norm,
etc.)

Open and closed sets, interior, closure, boundary

Convergence of sequences, completeness, compactness

Continuous functions, Lipschitz continuity, uniform continuity

Sequences of functions, pointwise convergence, uniform
convergence

Spaces of continuous functions, norms, metrics, functionals

Differentiation

Definition, mean value theorem, L’Hopital’s rule

Derivatives of higher order and Taylor’s theorem

Convexity and subdifferentials

Applications to Differential and Integral Equations

Fixed point theorem, method of successive approximations

Systems of linear equations, conditions for existence and
uniqueness of solutions (using different metrics)

Fredholm and Volterra equations, kernels, resolvants

Systems of differential equations, Picard’s method

Measure and Integration

Algebras, measurable spaces and functions

Measures, the Lebesgue measure, transformations of measures

Integrals, limit theorems

Riemann integrals, Stieltjes integrals, line and surface integrals

Change of variables, derivatives of measures, differentiation

Transition kernels, iterated integrals, measure-kernel-function
setup

Function Spaces

Normed vector spaces, linear operators and functionals

Hilbert spaces, orthogonality, projections

Fourier series

Legendre, Hermite, and Laguerre polynomials

Riesz representation theorem

Adjoint, Hermitian, unitary and normal operators

3 References

The following texts have been put on reserve in the Engineering library: